what is the formula between sensitivity and specificity?

Question

What is the formula between sensitivity and specificity ? Is it correct to say that it is : $$\operatorname{Sens} = 1- \operatorname{Spec}?$$
I know this question is a little silly but I couldn't find anything to help me.Thank you all in advance.

Answer 1

There's no direct mathematical relationship between sensitivity and specificity in general. You can read all the definitions on the wiki page, but to get an idea of why they need not be related you have to think about the definitions. Consider the case of a test for a disease.

Sensitivity or true positive rate tells you what proportion of people who have the disease will test positive.

Specificity or true negative rate tells you what proportion of people who don't have the disease will test negative.

Consider a perfect test: everyone who has the disease tests positive, everyone who doesn't tests negative. There are no false negatives or false positives. Both the sensitivity and the specificity are 1 - so it's certainly not the case that "sens = 1 - spec."

On the other hand, consider a test which always gives the answer positive. Sensitivity is 1: everyone who has the disease tests positive. Specificity is 0 because no-one tests negative. So you can have the same sensitivity and different specificities.

1 - specificity is the false negative rate, and 1 - sensitivity is the false positive rate.